Order 2 Quantum Wasserstein Distance Advances State Discrimination for Gaussian States

The challenge of quantifying the difference between quantum states has significant implications for fields ranging from machine learning to fundamental physics, and researchers continually seek more effective methods for this comparison. Anaelle Hertz from the National Research Council of Canada, Mohammad Ahmadpoor and Oleksandr Dzhenzherov from the University of Ottawa, alongside Khabat Heshami from the National Research Council of Canada and University of Calgary, now present a new tool for this task, a quantum Wasserstein distance specifically designed for Gaussian states. This achievement builds upon existing work in optimal transport, providing a general formula to calculate the minimal cost of transforming one Gaussian state into another, and importantly, allows for direct comparison with other established distance measures. By offering a closed-form solution for quantifying differences between these quantum states, this research establishes a powerful new approach for state discrimination and opens avenues for advancements in quantum information theory.

Gaussian States and Quantum Distance Measures

This document provides a comprehensive treatment of Gaussian states in quantum information theory, exploring their mathematical properties and various distance measures used to compare them. The work focuses on establishing relationships and bounds on these distances, assuming a strong background in quantum mechanics, linear algebra, probability theory, and information theory. The document reviews common quantum distance measures, including trace distance, fidelity, Bures distance, overlap, quantum relative entropy, and Hilbert-Schmidt distance. A key focus lies on proving inequalities, particularly relating the Bures and Wasserstein distances for thermal states.

The analysis centers on Gaussian states, allowing for tractable mathematical analysis and potential applications in quantum communication, cryptography, computation, state discrimination, and error correction. The work highlights the importance of inequalities in understanding relationships between distance measures and choosing the most appropriate measure for a given task. This method leverages the mathematical properties of optimal transport, a concept useful in fields like machine learning, to determine the minimal cost of transforming one quantum state into another. The resulting formula provides a precise measure of dissimilarity and allows for direct comparison with other established distance measures. The technique relies on characterizing quantum states using their Wigner functions, providing a phase-space representation analogous to classical probability distributions. By applying advanced mathematical tools, including concepts from matrix analysis and Douglas factorization, the team derived a formula that accurately quantifies the distance between Gaussian states. This formula recovers known results for classical Gaussian distributions and thermal states, validating its accuracy and broadening its applicability.

Wasserstein Distance for All Gaussian States

Scientists have achieved a general formula for calculating the Wasserstein distance between any two one-mode Gaussian states, expanding a previously established framework to a broader class of quantum states. This advancement allows for an operational interpretation based on physically implementable transformations. The resulting formula provides a precise mathematical relationship for determining the distance between these states. The research builds upon a formulation of quantum optimal transport, defining a coupling between states and minimizing the average cost associated with transforming one state into another.

By employing quantum channels, the team established a clear connection between the Wasserstein distance and physically realizable operations on quantum systems. The minimum cost, calculated through the derived formula, directly corresponds to the squared Wasserstein distance. The resulting formula provides a precise measure of dissimilarity and allows for direct comparison with other established distance measures. The formula accurately recovers previously known formulas for thermal states and classical Gaussian distributions. The work establishes a connection between the Wasserstein distance and quantum state similarity, offering a new perspective on comparing different measures. The research provides a foundation for future work extending the formula to multi-mode states and exploring its application to non-Gaussian states, potentially offering a way to quantify quantum features like nonclassicality.